标题： 全息理论中的纠缠谱 显示英文标题

标题： Entanglement spectra from holography

摘要： 纠缠谱的双量子系统由模哈密顿量的本征值分布给出。 在本工作中，我们计算了具有全息对偶的$d$维共形场理论（CFT）子区域的真空态中的纠缠谱。 在球面（或平面）纠缠表面的情况下，我们在二维情况下恢复了已知结果，包括高能区的 Cardy 公式。 在更高维度$d>2$中，我们解析地确定了一个在大能量下有效且与文献中 CFT 谱研究一致的 Cardy 公式的推广。 我们还数值地研究了远高于模基态能量的能级谱。 我们将分析扩展到爱因斯坦-麦克斯韦引力的超对称点，在$d=2,3$时提供了精确结果，并在一般维度$d$的高能情况下给出了 Cardy 公式的推广。 我们考虑了球面纠缠表面的小形状变形，包括非超对称和超对称情况。 在所有情况下，我们发现微正则熵与模能量的高能标度不受形状变形的影响。 这一结果表明，纠缠谱的高能区携带了与纠缠表面形状无关的普适信息。 显示英文摘要

摘要： The entanglement spectrum of a bipartite quantum system is given by the distribution of eigenvalues of the modular Hamiltonian. In this work, we compute the entanglement spectrum in the vacuum state for a subregion of a $d$-dimensional conformal field theory (CFT) admitting a holographic dual. In the case of a spherical (or planar) entangling surface, we recover known results in two dimensions, including the Cardy formula in the high energy regime. In higher dimensions $d>2$, we analytically determine a generalization of the Cardy formula valid at large energies and consistent with previous studies of CFT spectra in the literature. We also investigate numerically the spectrum at energy levels far above the modular ground state energy. We extend our analysis to the supersymmetric point of Einstein-Maxwell gravity, providing exact results when $d=2,3$, and a generalization of the Cardy formula at high energies in generic dimension $d$. We consider small shape deformations of a spherical entangling surface, for both the non-supersymmetric and the supersymmetric cases. In all cases we find that the high-energy scaling of the microcanonical entropy with the modular energy is unaffected by the shape deformation. This result suggests that the high-energy regime of the entanglement spectra carries universal information, independent of the shape of the entangling surface.